Victory at home to Scotland on Friday will guarantee qualification for the knockout stages with a game to play irrespective of what happens in the Group D match between Croatia and Czech Republic three hours earlier.
The Czechs lead the group after beating a Scotland team featuring Liam Cooper 2-0 at Hampden Park in their opening game yesterday.
Patrik Schick scored either side of half-time, his second from the halfway line to catch goalkeeper David Marshall way off his line.
Even two wins for Croatia in their remaining matches will not see them finish above England with places initially decided on head-to-head records.
The top two teams in Group D automatically qualify for the last 16, along with the six-group competition’s four best third-placed teams. The group winners play at Wembley in the next round, although on paper it would appear the draw could actually be easier if they finish second.
To have played as well as England did against such high-calibre opposition – albeit they only scored once, through Raheem Sterling – will raise expectations of a squad which was already highly-fancied. But Southgate is unconcerned.
“Pressure is what we decide it to be,” he argued.
“It’s a positive step in terms of qualification, but one of the hardest things in sport is to come back to that point you were at before the kick-off for the next game.
“You can get a bit of a glow of the win and a bit soft, a bit puddingy when you poke your finger in the belly.
“We’ve got to make sure that we’re ready for what is going to be an incredible game against Scotland and a huge challenge.
“You have to enjoy your wins, especially in tournaments, but focus now.
“Our first objective is qualification. We haven’t done that yet and we have a very important game for us and our fans on Friday night.”
Southgate also allayed fears about an ineffective performance from Harry Kane, who was substituted in Sunday’s second half.
“He’s been a reliable performer for us for a long time, and his goals record in the last two-and-a-half years, with us and his club, is excellent,” he pointed out. “Even this year, when people have been saying he’s not hit the heights, he’s still one in three for his club. We can’t have all the burden of scoring on Harry’s shoulders – you’ve got to have players in those wide positions and in midfield who can chip in as well.”
Scotland were yesterday without one of their key players, left-sided centre-back Kieran Tierney, with a minor injury not expected to keep him out at Wembley.
“Kieran picked up a little niggle in training through the week,” manager Steve Clarke confirmed. “Hopefully (he’ll be) ready for the next one.”
Sheffield United midfielder John Fleck was left out of Scotland’s matchday squad having missed 10 days of build-up with Covid-19.